Near-Optimality of Linear Recovery in Gaussian Observation Scheme under $\|\cdot\|_2^2$-Loss

TL;DR

This paper demonstrates that in Gaussian observation schemes, simple linear estimators are nearly optimal for recovering linear images of signals from convex sets, without restrictions on the observation or transformation matrices.

Contribution

It establishes near-optimality of linear recovery methods under broad conditions, extending previous results to general matrices A and B and convex sets X.

Findings

Linear estimates are near-optimal in Gaussian noise settings.

No restrictions on matrices A and B are needed for the results.

Applicable to convex sets like intersections of ellipsoids and cylinders.

Abstract

We consider the problem of recovering linear image $Bx$ of a signal $x$ known to belong to a given convex compact set $X$ from indirect observation $\omega=Ax+\sigma\xi$ of $x$ corrupted by Gaussian noise $\xi$. It is shown that under some assumptions on $X$ (satisfied, e.g., when $X$ is the intersection of $K$ concentric ellipsoids/elliptic cylinders), an easy-to-compute linear estimate is near-optimal, in certain precise sense, in terms of its worst-case, over $x\in X$, expected $\|\cdot\|_2^2$-error. The main novelty here is that our results impose no restrictions on $A$ and $B$, to the best of our knowledge, preceding results on optimality of linear estimates dealt either with the case of direct observations $A=I$ and $B=I$, or with the "diagonal case" where $A$, $B$ are diagonal and $X$ is given by a "separable" constraint like $X=\{x:\sum_ia_i^2x_i^2\leq 1\}$ or $X=\{x:\max_i|a_ix_i|\leq1\}$, or with estimating a linear form (i.e., the case one-dimensional $Bx$).